Talk:Beeman's algorithm

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Equations

The second equation doesnot make sense $v(t+\Delta t)=v(t)+v(t)\Delta t+...$ changes in velocity could not be proportional to velocity itself! abakharev 08:04, 15 February 2006 (UTC)[reply]

It looks like the equation was made by someone substituting one of the predictor-corrector modifications and removing the predictor-corrector terms. I removed the reference from the page that links to the incorrect source, and added one with a more complete list. I do not know if the error term is correct, but the source I added seems much more reliable, so I suspect the error term from that source is also correct. Mattopia 08:55, 15 February 2006 (UTC)[reply]

At least it makes sense now. Thanks a lot abakharev 09:01, 15 February 2006 (UTC)[reply]

I was unable to find the original paper from Beeman on it. That second equation looked a little screwy to me to, but I didn't know it well enough to second guess it properly. --Numsgil 12:47, 15 February 2006 (UTC)[reply]

Error term

The first source says the error term on velocity is O(dt^4), whereas the other two sources claim O(dt^3). The best step is to probably find Beeman's original paper on the algorithm and be certain one way or another. --Numsgil 13:02, 15 February 2006 (UTC)[reply]

The errors of derivative are almost universally one order worse than the errors of the main value. So I will be surprised if the order would be dt^4 for both position and velocity. I took the liberty to edit the formula, it would be still worth to find the original paper anyway abakharev 16:34, 15 February 2006 (UTC)[reply]

I think it's O(dt^3) also, however velocity verlet is an example of an algorithm with the same order of error for position and velocity. --Numsgil 19:11, 15 February 2006 (UTC)[reply]

Bad link

Basic Molecular Dynamics - page and pdf file were deleted. --Vadikus (talk) 10:51, 21 October 2008 (UTC)[reply]

Beeman, 1976

I've put a citation for the original Beeman paper into the source (hidden in a comment). If someone who has access to J. Comp. Phys. could please have a look at it and confirm that it is an appropriate reference, then that should save a little bit of time putting the details together! --Philtweir (talk) 12:34, 16 June 2010 (UTC)[reply]

The equations of the order 3 method in the paper are

{\begin{aligned}{\text{predictor }}&\\r_{n+1}&=r_{n}+hv_{n}+{\frac {h^{2}}{6}}(4a_{n}-a_{n-1})+{\frac {h^{4}}{8}}r_{n}^{(4)};\\{\text{corrector }}&{\text{ (after computation of }}a_{n+1})\\r_{n+1}&=r_{n}+hv_{n}+{\frac {h^{2}}{6}}(a_{n+1}+2a_{n})-{\frac {h^{4}}{12}}r_{n}^{(4)};\\hv_{n+1}&=r_{n+1}-r_{n}+{\frac {h^{2}}{6}}(2a_{n+1}+a_{n})-{\frac {h^{4}}{24}}r_{n}^{(4)}.\end{aligned}}

Additionally, methods of order 4 and 5 are given and numerically compared to Verlet, Adams-Multon-Bashford multistep, Rahman and Nordsiek methods. Verlet is not used in the leapfrog or velocity Verlet variants, in this situation Beeman's third order method appeared more stable for larger steplengths. The x steps are the same as in the article, the corrector steps are at odds.--LutzL (talk) 14:52, 6 September 2010 (UTC)[reply]